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No. Depositions are different than trials. The right of cross examination exists in trials. Depositions are discovery mechanisms. Pretty much any question is allowed in a deposition (except for attorney client privileged or irrelevant questions). The other client doesn't have to be present (and most of the time isn't, as they would have nothing to do in the first place).
This is exactly how depositions go. A deposition is not a trial. The reporter is not a judge or "running" the deposition. So there's nothing that happened wrong here. You could do the same thing for the other party. This is how depositions go.
Furthermore, "cross examination" is where an adverse attorney addresses you, on the stand, after direct examination is complete (by your attorney). This is in court, not a deposition. So there's no cross examination right implicated at all here as it was simply a deposition.
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